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تسجيل مستخدم جديدLow chromatic spanning sub(di)graphs with prescribed degree or connectivity properties
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Generalizing well-known results of ErdH{o}s and Lovasz, we show that every graph $G$ contains a spanning $k$-partite subgraph $H$ with $lambda{}(H)geq lceil{}frac{k-1}{k}lambda{}(G)rceil$, where $lambda{}(G)$ is the edge-connectivity of $G$. In particular, together with a well-known result due to Nash-Williams and Tutte, this implies that every $7$-edge-connected graphs contains a spanning bipartite graph whose edge set decomposes into two edge-disjoint spanning trees. We show that this is best possible as it does not hold for infintely many $6$-edge-connected graphs. For directed graphs, it was shown in [6] that there is no $k$ such that every $k$-arc-connected digraph has a spanning strong bipartite subdigraph. We prove that every strong digraph has a spanning strong 3-partite subdigraph and that every strong semicomplete digraph on at least 6 vertices contains a spanning strong bipartite subdigraph. jbj{We generalize this result to higher connectivities by proving} that, for every positive integer $k$, every $k$-arc-connected digraph contains a spanning $(2k+1$)-partite subdigraph which is $k$-arc-connected and this is best possible. A conjecture in [18] implies that every digraph of minimum out-degree $2k-1$ contains a spanning $3$-partite subdigraph with minimum out-degree at least $k$. We prove that the bound $2k-1$ would be best possible by providing an infinite class of digraphs with minimum out-degree $2k-2$ which do not contain any spanning $3$-partite subdigraph in which all out-degrees are at least $k$. We also prove that every digraph of minimum semi-degree at least $3r$ contains a spanning $6$-partite subdigraph in which every vertex has in- and out-degree at least $r$.
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